The bending elasticity of the monoglyceride 1-monoolein (IMO) in water has been measured in the inverse hexagonal(HII) phase at 37 degrees C. At this temperature, fully hydrated MO is normally in an inverse bicontinuous cubic phase based on Schwarz's D surface, Q(II)(D);. The addition of either of the C-23 chainlength hydrocarbons, tricosane or 9-cis-tricosene, at mole fractions with respect to MO in excess of 0.05, induces a phase transition into the H-II phase. This transition is understood to occur because packing stresses in the hydrophobic regions of the! H-II phase are reduced to levels where this phase is at a lower free energy than the Q(II)(D) phase. We have used X-ray diffraction on gravimetrically prepared samples and samples subjected to an osmotic stress to determine the bending energy of MO in this phase. The evidence suggests that 9-cis-tricosene can relieve almost all of the packing stress in the Hn phase. In this case, we fmd that the spontaneous radius of curvature at the pivotal surface, R-0, is -20.0 +/-. 0.3 Angstrom, and the monolayer bending rigidity, kappa, is (1.2 +/- 0.1) x 10(-20) J. Comparing these energetics with those of the Q(II)(D) phase indicates that the packing stress for an excess water, H-II phase at 37 degrees C in the absence of 9-cis-tricosene would constitute at least 50% of the total equilibrium free energy. With tricosane, the packing stress cannot be completely relieved because tricosane melts at 47.6 degrees C in the bulk. The results from geometric measurements on this system suggest that when the packing stress is not fully relieved in the H-II phase, the polar/apolar interface is deformed away from being cylindrical. Treating the interface as if it were cylindrical leads to a nonphysical location of the pivotal surface that is in disagreement with all previous measurements, a 35% increase in the magnitude of R-0, and a 4-fold increase in the calculated bending rigidity.